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Standard methodologies for the extraction of the stochastic gravitationalwave background (SGWB) from auto- or cross-correlation of interferometricsignals often involve the use of a filter function. The standard optimal filtermaximizes the signal-to-noise ratio (SNR) between the total SGWB and the noise.We derive expressions for the optimal filter and SNR in the presence of atarget SGWB plus other unwanted components. We also generalize the methodologyto the case of template-free reconstruction. The formalism allows to easilyperform analyses and forecasts that marginalize over foreground signals, suchas the typical $\Omega_{\rm GW} \propto f^{2/3}$ background arising from binarycoalescence. We demonstrate the methodology with the LISA mission and discusspossible extensions and domains of application.

Measuring the primordial gravitational wave background in the presence of other stochastic signals